The logarithmic number system has well known distinct advantages over other number systems including a wide range of number representations in a short and easily representable format. Signal processors which must process many multiplication operations may realize significant speed and size advantages by using the logarithmic form of multiplication. A disadvantage of the logarithmic number system has previously been the difficulty associated with implementing addition operations. Due to the nonlinear nature of logarithmic addition and subtraction, logarithmic circuits have not been commonly used because addition and subtraction is difficult to implement.
A technique used to implement a logarithmic addition or subtraction is to add a predetermined correction factor to the operand having the minimum exponential value. The correction factors are typically stored in memory circuits such as ROMs or programmable logic arrays (PLAs). An obvious disadvantage with such an implementation is the large amount of memory required to store a sufficient number of correction factors to maintain usable signal to quantization error ratios while minimizing harmonic distortions. The amount of memory needed is directly proportional to the amount of precision required. Precision is affected by two factors. The first precision factor is associated with the total number of memory locations implemented. An increase in the number of locations in the memory tables increases the precision of the addition operations.
The second precision factor is associated with the number of bits used to represent each of the correction factors. Therefore, precision of a logarithmic adder is directly proportional to the size of the memory table. A technique to reduce the number of correction factors needed for storage in memory is proposed by M. L. Frey and F. J. Taylor in "A Table Reduction Technique for Logarithmically Architected Digital Filters" in the IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-33, No. 3, June 1985, pages 718-719. Frey et al. group correction factors to minimize the number of memory locations and further propose an algorithm for determining endpoint values of the groups. A disadvantage of this technique is that an additional quantization error is associated with almost every calculation. A further disadvantage of the grouping technique taught by Frey et al. is the large number of bits required to represent the correction factors.